# Related Structures

In this section functions for creating other structures from a root datum are briefly listed. See the appropriate chapters of the Handbook for more details.

## `RootSystem(R): RootDtm -> RootSys`

The root system corresponding to the root datum $R$. See Chapter [Root Systems](../RootSystems/index-root-systems.md#chaprootsys).

## `CoxeterGroup(grpcat, R): Cat, RootDtm -> grpcat`

The Coxeter group (of type `grpcat`) of a root datum $R$. There are variations of this signature. The first argument can be `GrpMat`, `GrpPermCox`, `GrpPerm`, `GrpFPCox` or `GrpFP` and the second argument can be a root system or root datum. (See Chapter [Coxeter Groups](../CoxeterGroups/index-coxeter-groups.md#chapgrpcox).) If the first argument is `GrpFPCox` the braid group and pure braid group can be computed from the Coxeter group using the commands in Section [Braid Groups](../CoxeterGroups/group-braid.md#sectgrpcoxbraid).

## `CoxeterGroup(R): RootDtm -> GrpPermCox`

## `WeylGroup(R): RootDtm -> GrpPermCox`

The permutation Coxeter group with root datum $R$. See Chapter ChapGrpPermCox.

## `CoxeterGroup(GrpPermCox, R): Cat, RootDtm -> GrpPermCox`

## `ReflectionGroup(R): RootDtm -> GrpMat`

The reflection group of the root datum $R$. See Chapter [Reflection Groups](../ReflectionGroups/index-reflection-groups.md#chapgrprfl).

## `LieAlgebraHomorphism(phi, k): Map, Rng -> AlgLie`

The homomorphism of reductive Lie algebras over the ring $k$ corresponding to the root datum morphism $\phi$. See Chapter [Lie Algebras](../LieAlgebras/index-lie-algebras.md#chapalglie).

## `LieAlgebra(R, k): RootDtm, Rng -> AlgLie`

The reductive Lie algebra over the ring $k$ with root datum $R$. See Chapter [Lie Algebras](../LieAlgebras/index-lie-algebras.md#chapalglie).

## `GroupOfLieType(R, k): RootDtm, Rng -> GrpLie`

The group of Lie type over the ring $k$ with root datum $R$. See Chapter [Groups of Lie Type](../ChevalleyGroups/index-chevalley-groups.md#chapgrplie).

## `GroupOfLieTypeHomomorphism(phi, k): Map, Rng -> GrpLie`

The algebraic homomorphism of groups of Lie type over the ring $k$ corresponding to the root datum morphism $\phi$. See Chapter [Groups of Lie Type](../ChevalleyGroups/index-chevalley-groups.md#chapgrplie).

## `Example: Related (ex-174972)`

```magma
> R := RootDatum("b3");
> SemisimpleType(LieAlgebra(R, Rationals()));
B3
> #CoxeterGroup(R);
48
> GroupOfLieType(R, Rationals());
$: Group of Lie type B3 over Rational Field

```